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Theorem addid1 8992
Description:  0 is an additive identity. This used to be one of our complex number axioms, until it was found to be dependent on the others. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.) (Proof shortened by Mario Carneiro, 27-May-2016.)
Assertion
Ref Expression
addid1  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )

Proof of Theorem addid1
Dummy variables  c  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1re 8837 . 2  |-  1  e.  RR
2 ax-rnegex 8808 . 2  |-  ( 1  e.  RR  ->  E. c  e.  RR  ( 1  +  c )  =  0 )
3 ax-1ne0 8806 . . . . . 6  |-  1  =/=  0
4 oveq2 5866 . . . . . . . . . 10  |-  ( c  =  0  ->  (
1  +  c )  =  ( 1  +  0 ) )
54eqeq1d 2291 . . . . . . . . 9  |-  ( c  =  0  ->  (
( 1  +  c )  =  0  <->  (
1  +  0 )  =  0 ) )
65biimpcd 215 . . . . . . . 8  |-  ( ( 1  +  c )  =  0  ->  (
c  =  0  -> 
( 1  +  0 )  =  0 ) )
7 oveq2 5866 . . . . . . . . 9  |-  ( ( 1  +  0 )  =  0  ->  (
( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )
8 ax-icn 8796 . . . . . . . . . . . . . . 15  |-  _i  e.  CC
98, 8mulcli 8842 . . . . . . . . . . . . . 14  |-  ( _i  x.  _i )  e.  CC
109, 9mulcli 8842 . . . . . . . . . . . . 13  |-  ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  e.  CC
11 ax-1cn 8795 . . . . . . . . . . . . 13  |-  1  e.  CC
12 0cn 8831 . . . . . . . . . . . . 13  |-  0  e.  CC
1310, 11, 12adddii 8847 . . . . . . . . . . . 12  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  ( 1  +  0 ) )  =  ( ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  1 )  +  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )
1410mulid1i 8839 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  1 )  =  ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )
15 mul01 8991 . . . . . . . . . . . . . . 15  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  e.  CC  ->  ( (
( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  0 )
1610, 15ax-mp 8 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  0
17 ax-i2m1 8805 . . . . . . . . . . . . . 14  |-  ( ( _i  x.  _i )  +  1 )  =  0
1816, 17eqtr4i 2306 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  =  ( ( _i  x.  _i )  +  1 )
1914, 18oveq12i 5870 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  1 )  +  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 ) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )
2013, 19eqtri 2303 . . . . . . . . . . 11  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )
2120, 16eqeq12i 2296 . . . . . . . . . 10  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  <->  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1 ) )  =  0 )
2210, 9, 11addassi 8845 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )  +  1 )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1 ) )
239mulid1i 8839 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  1 )  =  ( _i  x.  _i )
2423oveq2i 5869 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1
) )  =  ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )
259, 9, 11adddii 8847 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1 ) )
2617oveq2i 5869 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  ( ( _i  x.  _i )  x.  0
)
27 mul01 8991 . . . . . . . . . . . . . . . . 17  |-  ( ( _i  x.  _i )  e.  CC  ->  (
( _i  x.  _i )  x.  0 )  =  0 )
289, 27ax-mp 8 . . . . . . . . . . . . . . . 16  |-  ( ( _i  x.  _i )  x.  0 )  =  0
2926, 28eqtri 2303 . . . . . . . . . . . . . . 15  |-  ( ( _i  x.  _i )  x.  ( ( _i  x.  _i )  +  1 ) )  =  0
3025, 29eqtr3i 2305 . . . . . . . . . . . . . 14  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  x.  1
) )  =  0
3124, 30eqtr3i 2305 . . . . . . . . . . . . 13  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( _i  x.  _i ) )  =  0
3231oveq1i 5868 . . . . . . . . . . . 12  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( _i  x.  _i ) )  +  1 )  =  ( 0  +  1 )
3322, 32eqtr3i 2305 . . . . . . . . . . 11  |-  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  +  ( ( _i  x.  _i )  +  1
) )  =  ( 0  +  1 )
34 00id 8987 . . . . . . . . . . . 12  |-  ( 0  +  0 )  =  0
3534eqcomi 2287 . . . . . . . . . . 11  |-  0  =  ( 0  +  0 )
3633, 35eqeq12i 2296 . . . . . . . . . 10  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  +  ( ( _i  x.  _i )  +  1 ) )  =  0  <->  ( 0  +  1 )  =  ( 0  +  0 ) )
37 0re 8838 . . . . . . . . . . 11  |-  0  e.  RR
38 readdcan 8986 . . . . . . . . . . 11  |-  ( ( 1  e.  RR  /\  0  e.  RR  /\  0  e.  RR )  ->  (
( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 ) )
391, 37, 37, 38mp3an 1277 . . . . . . . . . 10  |-  ( ( 0  +  1 )  =  ( 0  +  0 )  <->  1  = 
0 )
4021, 36, 393bitri 262 . . . . . . . . 9  |-  ( ( ( ( _i  x.  _i )  x.  (
_i  x.  _i )
)  x.  ( 1  +  0 ) )  =  ( ( ( _i  x.  _i )  x.  ( _i  x.  _i ) )  x.  0 )  <->  1  =  0 )
417, 40sylib 188 . . . . . . . 8  |-  ( ( 1  +  0 )  =  0  ->  1  =  0 )
426, 41syl6 29 . . . . . . 7  |-  ( ( 1  +  c )  =  0  ->  (
c  =  0  -> 
1  =  0 ) )
4342necon3d 2484 . . . . . 6  |-  ( ( 1  +  c )  =  0  ->  (
1  =/=  0  -> 
c  =/=  0 ) )
443, 43mpi 16 . . . . 5  |-  ( ( 1  +  c )  =  0  ->  c  =/=  0 )
45 ax-rrecex 8809 . . . . 5  |-  ( ( c  e.  RR  /\  c  =/=  0 )  ->  E. x  e.  RR  ( c  x.  x
)  =  1 )
4644, 45sylan2 460 . . . 4  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  E. x  e.  RR  ( c  x.  x
)  =  1 )
47 simpr 447 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  A  e.  CC )
48 simplrl 736 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  x  e.  RR )
4948recnd 8861 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  x  e.  CC )
5047, 49mulcld 8855 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  ( A  x.  x )  e.  CC )
51 simplll 734 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  c  e.  RR )
5251recnd 8861 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  c  e.  CC )
5312a1i 10 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  0  e.  CC )
5450, 52, 53adddid 8859 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  ( c  +  0 ) )  =  ( ( ( A  x.  x )  x.  c )  +  ( ( A  x.  x )  x.  0 ) ) )
5511a1i 10 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  1  e.  CC )
5655, 52, 53addassd 8857 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  c )  +  0 )  =  ( 1  +  ( c  +  0 ) ) )
57 simpllr 735 . . . . . . . . . . . . 13  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  c )  =  0 )
5857oveq1d 5873 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  c )  +  0 )  =  ( 0  +  0 ) )
5956, 58eqtr3d 2317 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  ( c  +  0 ) )  =  ( 0  +  0 ) )
6034, 59, 573eqtr4a 2341 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  +  ( c  +  0 ) )  =  ( 1  +  c ) )
6137a1i 10 . . . . . . . . . . . 12  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  0  e.  RR )
6251, 61readdcld 8862 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  +  0 )  e.  RR )
631a1i 10 . . . . . . . . . . 11  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  1  e.  RR )
64 readdcan 8986 . . . . . . . . . . 11  |-  ( ( ( c  +  0 )  e.  RR  /\  c  e.  RR  /\  1  e.  RR )  ->  (
( 1  +  ( c  +  0 ) )  =  ( 1  +  c )  <->  ( c  +  0 )  =  c ) )
6562, 51, 63, 64syl3anc 1182 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( 1  +  ( c  +  0 ) )  =  ( 1  +  c )  <->  ( c  +  0 )  =  c ) )
6660, 65mpbid 201 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  +  0 )  =  c )
6766oveq2d 5874 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  ( c  +  0 ) )  =  ( ( A  x.  x )  x.  c ) )
6854, 67eqtr3d 2317 . . . . . . 7  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( ( A  x.  x )  x.  c
)  +  ( ( A  x.  x )  x.  0 ) )  =  ( ( A  x.  x )  x.  c ) )
69 mul31 8980 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  x  e.  CC  /\  c  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  ( ( c  x.  x )  x.  A ) )
7047, 49, 52, 69syl3anc 1182 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  ( ( c  x.  x )  x.  A ) )
71 simplrr 737 . . . . . . . . . 10  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
c  x.  x )  =  1 )
7271oveq1d 5873 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( c  x.  x
)  x.  A )  =  ( 1  x.  A ) )
7347mulid2d 8853 . . . . . . . . 9  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
1  x.  A )  =  A )
7470, 72, 733eqtrd 2319 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  c )  =  A )
75 mul01 8991 . . . . . . . . 9  |-  ( ( A  x.  x )  e.  CC  ->  (
( A  x.  x
)  x.  0 )  =  0 )
7650, 75syl 15 . . . . . . . 8  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( A  x.  x
)  x.  0 )  =  0 )
7774, 76oveq12d 5876 . . . . . . 7  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  (
( ( A  x.  x )  x.  c
)  +  ( ( A  x.  x )  x.  0 ) )  =  ( A  + 
0 ) )
7868, 77, 743eqtr3d 2323 . . . . . 6  |-  ( ( ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  /\  (
x  e.  RR  /\  ( c  x.  x
)  =  1 ) )  /\  A  e.  CC )  ->  ( A  +  0 )  =  A )
7978exp42 594 . . . . 5  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( x  e.  RR  ->  ( (
c  x.  x )  =  1  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) ) ) )
8079rexlimdv 2666 . . . 4  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( E. x  e.  RR  ( c  x.  x )  =  1  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) ) )
8146, 80mpd 14 . . 3  |-  ( ( c  e.  RR  /\  ( 1  +  c )  =  0 )  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) )
8281rexlimiva 2662 . 2  |-  ( E. c  e.  RR  (
1  +  c )  =  0  ->  ( A  e.  CC  ->  ( A  +  0 )  =  A ) )
831, 2, 82mp2b 9 1  |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544  (class class class)co 5858   CCcc 8735   RRcr 8736   0cc0 8737   1c1 8738   _ici 8739    + caddc 8740    x. cmul 8742
This theorem is referenced by:  cnegex  8993  addid2  8995  addcan2  8997  addid1i  8999  addid1d  9012  subid  9067  subid1  9068  shftval3  11571  reim0  11603  isercolllem3  12140  fsumcvg  12185  summolem2a  12188  ovolicc1  18875  relexpadd  24035  brbtwn2  24533  axsegconlem1  24545  ax5seglem4  24560  axeuclid  24591  axcontlem2  24593  axcontlem4  24595  stoweidlem11  27760  stoweidlem26  27775
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-ltxr 8872
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